Fast Time-Varying Multipath Channel Estimation and Signal Detection for of OFDM Systems with Interference Compensation

Abstract

Orthogonal frequency division multiplexing (OFDM) is a widely adopted system for broadband wireless transmission, However, time-varying channels degrade the performance owing to the intercarrier interference (ICI). This study discusses the performance enhancement problems of channel estimation and signal detection in view of fast time-varying multipath fading channel characteristics of high mobility. A two-stage method is presented. In the coarse stage, initial channel estimation is performed using the signals at pilot subcarriers based on least-square (LS) estimate is performed. However, the initial estimate still needs to be improved because of the data-induced ICI. The coarse stage improves estimate accuracy by compensating the interferences. The refining stage, applies interference compensation and q-banded successive interference suppression (qSIS) methods to enhance the detection and re-estimate accuracy. Based on robustness, the detected data are called the “pseudo-pilots”, which together with the pilots are treated as a “visual-training” (VT) to improve channel re-estimation. Simulation results indicate that the proposed method is more robust than the traditional method in fast time-varying multipath fading channels with lower computational complexity. The proposed method is independent of the delay and fading properties of the channel.

Appendix A: q-Banded Successive Interference Suppression

Let H _D and Y _D be taken from the data subcarriers of H _r and Y, respectively. The relation of H _D and H _r is H _D (m,n) = H _r (a,b) with a = κD + r + 1 and b = λD + ω + 1, in which κ = Int(m/(D − 1)); λ = Int(n/(D − 1)); r = (m) _D − 1; ω = (n) _D − 1; m = 0, 1, …, N – Np − 1; n = 0, 1, …, N − Np − 1. Similarly, for Y _D (m) = Y(a), we have a = κD + r + 1, where κ = Int(m/(D − 1)), r = (m) _D − 1, and m = 0, 1, …, N − Np − 1.

For convenience, the MATLAB-style notation A(i:j,k:l) is adopted to represent a submatrix of A with rows i through j and columns k through l. Furthermore, we use A(i,:) and A(:,j) to indicate the ith row and the jth column of A respectively. The term A(i:j,k) represents the column vector of A that comprising rows i through j at kth column of A, and likewise for the row vector A(k,i:j).

The qSIS method performs an iterative detection that initializes the detection process by identifying the subcarrier with strongest power first and leaving the remaining subcarrier to be detected in successive iterations. A list of detection sequence is required. This study employs subcarrier power to quantify the subcarrier quality. Let the power vector P _D be denoted as:

$$P_{D} (n) = \sum\limits_{m = 0}^{{N - N_{p} - 1}} {\left\| {H_{D} (m,n)} \right\|^{2} } ,\;\;n = 0,1, \ldots ,\;\;N - N_{p} - 1$$

(28)

Figure 14 depicts the flowchart of the proposed qSIS method. The ordered sequence K _D is obtained from the sorted P _D , where K _D (i) = j means that the j-th subcarrier will be detected at the i-th iteration. For simplification, the superscript ⁱ indicates the detection at the i-th iteration, marked by X ⁱ _D , from H ⁱ _D and Y ⁱ _D . For initialization, set i = 0, H ⁰ _D  = H _D , and Y ⁰ _D  = Y _D . Considering only q− banded dominant ICI terms in H ⁱ _D , at the i-th iteration, the j-th subcarrier has the largest power, and its signal of this subcarrier can be estimated as follows:

$${\mathbf{X}}_{D}^{i} (j) = Q[({\mathbf{A}}_{{}}^{i} )_{r}^{ + } {\mathbf{Y}}_{q}^{i} ]$$

(29)

where (A ⁱ) ⁺ _r indicates the r- th row of (A ⁱ)⁺, and Y ⁱ _q  = Y ⁱ _D (m:n)∈ ℂ^{(2q + 1) × 1} and A ⁱ = H ⁱ _D (m:n,m:n)∈ ℂ^{(2q + 1)×(2q + 1)}, with

$$m = \left\{ {\begin{array}{*{20}l} {0,} \hfill \\ {N - N_{p} - 1 - 2q,} \hfill \\ {j - q,} \hfill \\ \end{array} } \right. \, \begin{array}{*{20}l} {j - q \le 0} \hfill \\ { \, j + q > N - N_{p} - 1} \hfill \\ {else} \hfill \\ \end{array}$$

(30)

$$n = \left\{ {\begin{array}{*{20}l} {2q,} \hfill \\ {N - N_{p} - 1,} \hfill \\ {j + q,} \hfill \\ \end{array} } \right. \, \begin{array}{*{20}l} {j - q \le 0} \hfill \\ { \, j + q > N - N_{p} - 1} \hfill \\ {else} \hfill \\ \end{array}$$

(31)

The estimated j-th subcarrier can then be rewritten into

$$r = \left\{ {\begin{array}{*{20}l} {j,} \hfill \\ {j - (N - Np - 1 - 2q),} \hfill \\ {q + 1,} \hfill \\ \end{array} } \right. \, \begin{array}{*{20}l} {j - q \le 0} \hfill \\ { \, j + q > N - N_{p} - 1} \hfill \\ {else} \hfill \\ \end{array}$$

(32)

After the j-th subcarrier is determined, the interference compensation is obtained as

$${\mathbf{Y}}_{D}^{i + 1} = {\mathbf{Y}}_{D}^{i} - {\mathbf{H}}_{D}^{i} (:,j){\hat{\mathbf{X}}}_{D}^{i} (j)$$

(33)

Assuming that the j-th subcarrier interference has been eliminated in (33), the j-th column of channel matrix H ⁱ _D should be updated by nulling the corresponding column. This iterative procedure should be performed until the signal of the last subcarrier is determined. The detailed operation of the presented qSIS algorithm is summarized as follows:

• (0) Set the initial channel matrix H ⁰ _D  = H _D and Y ⁰ _D  = Y _D ; set i = 0.

• (1) Determine the power vector P _D to obtain an ordered sequence K _D .

• (2) For j = K _D (i), obtain A ⁱ and Y ⁱ _q from H ⁱ _D and Y ⁱ _D , respectively.

• (3) Obtain X ⁱ _D (j) from (29)

• (4) Calculate Y ⁱ⁺¹ _D from (33)

• (5) If i = N − N _p  − 1, then end the detection; otherwise, update the channel matrix H ⁱ⁺¹ _D by nulling the j-th column; then i = i + 1and go to step 2).

Finally, the qSIS output is given by:

$$X_{d}^{{(2)}} (k) = \left\{ {\begin{array}{*{20}l} {X_{D}^{i} (m),} &; {if{\text{ }}k = Int(\frac{m}{{D - 1}}) \times D + \bmod (m,D - 1) + 1{\text{ }}and{\text{ }}i = N - N_{p} - 1} \\ 0 &; {else} \\ \end{array} } \right.$$

(34)

where X ⁽²⁾ _d ∈ ℂ^N × 1, k∈ Ω _d , and m = 0,1,…, N− N _p  − 1.

Table 3 lists the signal detection for H _D and the complexity comparison from ZF, SIS and the proposed qSIS is presented. As discussed in [26], the main drawback of conventional SIS detection is the iterative process. This study found that the conventional needs to perform matrix inversions of O((N − N _p )^2.376) for total (N − N _p ) times. The computational complexity of conventional SIS is still impractical for real time systems, particularly with a large number of data subcarriers N − N _p . For qSIS, the size of partial matrix A ⁱ in (29) is (2q + 1) × (2q + 1) reducing the required inverse operation to estimate j-th subcarrier’s signal is O((2q + 1)^2.376). The qSIS detection is the iterative process that need to perform matrix inversions of O(N ^2.376 ) for total N − N _p times. Hence the complexity of the proposed qSIS is only (N − N _p )(2q + 1)^2.376. Figure 15 illustrates the flowchart of the proposed low- complexity qSIS method. ZF- only requires one matrix inverse operation with computation complexity of O((N − N _p )^2.376) [1] [27]. Therefore, the complexity gap between qSIS and SIS becomes more significant with larger N − N _p . For case N = 512 and N _p  = 256, SIS is 673 times as complex as ZF. The qSIS (q = 5) only requires 14.4% of ZF complexity, and 0.021% of SIS complexity. The qSIS with q = 7 requires 30% of ZF complexity, and 0.045% of SIC complexity, while qSIS with q = 9 requires 0.079% of SIS complexity and is only 53% more complex than ZF.

Table 3 Comparison of computational complexity for ZF, SIC, and the proposed qSIS, using N = 512 and N _p  = 256 with various q values

Fig. 15

The flowchart of the proposed low complexity qSIS method